Stability compensation of an admittance-controlled cartesian robot considering physical interaction with a human operator

2.1. Admittance control

The admittance control is a way to control robots with human interaction; control input to the system will come from the applied force of human operators, and the robot will move according to the admittance model to the desired position. Admittance is the inverse of impedance, and an impedance model is a transfer function that receives position information and calculates force output; therefore, an admittance model will receive force input information, $$ f_h $$, and calculate position information, $$ x_r $$. In combination with a position-controlled robot, as shown in Figure 1, where $$ G $$ represents the admittance model, and $$ G_r $$ represents the robot characteristic transfer function. The calculated position reference, $$ x_r $$, can be directly fed into the system, and the robot position, $$ x $$, will move to the desired position.

Figure 1. Block diagram of an admittance control robot, where $$ G $$ represents the admittance model, and $$ G_r $$ represents the robot characteristic transfer function.

We consider the mass spring damper system as an admittance model. However, in our study, it does not need to return to the origin after the operator breaks the contact, so the stiffness parameter can be neglected. The transfer function of the admittance model, $$ G $$, will be represented as Equation (1). Lowering the coefficient results in a lighter workload for the operator and less stability in the system. On the other hand, a higher coefficient increases the workload for the human operator in exchange for greater system stability.

2.2. Human-machine system

The robot requires instructions to move according to the needs of the human operator. The admittance control method is used when control is input via human-robot interaction. As shown in Figure 2, a robot that makes contact with the operator forms a single-couple system, and human interaction creates a closed-loop system^[5]. The impedance of the human operator, $$ H $$, will create force, $$ f_h $$, according to errors that come from the displacement of the robot, $$ x $$, subtracted from the intended position of the operator, $$ x_d $$. The force that a human operator needs to apply to the system is highly dependent on the admittance characteristics of the system.

Figure 2. Human-machine interaction diagram.

Closed-loop systems can have instability problems that also depend on $$ G_r $$, which represents the transfer function of the robot. System instability will occur if it is not optimally implemented, particularly because the dead time that exists in every system will cause a system delay in responding to commands and insufficient stiffness of the structure, resulting in oscillation if the admittance model is not low enough. However, if the admittance model is too low, it will affect maneuverability. So, we will derive the model of the experiment system that takes into account these properties mentioned above. Furthermore, we also consider a compensator for implementation in the system.

3.1. Experiment setup

The experiment setup shown in Figure 3 is designed for a power-assisted robot system. It is equipped with a force sensor, AC servo motors, servo packs, and linear sliders. A laser distance sensor is used for only spectrum analysis, and an acceleration sensor is used for feedback compensation mentioned later. These devices are connected to a PC, as shown in Figure 4, to form the experiment setup, where the PC performs measurements, calculates control operations, and outputs position command pulses to the servo packs with a sampling time of 1 ms.

Figure 3. Experiment setup.

Figure 4. Experiment setup diagram.

3.2. Modeling

3.2.2. Structure transfer function

From the test bench shown in Figure 3, a structure has an X- and Y-axis actuator in the system, as shown in Figure 5. From our previous research, we found that the X-axis is more unstable compared to the Y-axis when a human operator interacts with the robot. So, in this study, we will concentrate on only the X-axis, which leads to system instability. We consider the horizontal structure movement as a single mass spring damper system in which the parameters are $$ m_f $$, $$ k_f $$, and $$ d_f $$. For the X-axis actuator, the actuation force is displayed as $$ u $$, and the viscosity of the actuator is displayed as $$ d_e $$. It applies force to the Y-axis actuator and the frame. The mass of the Y-axis actuator is displayed as $$ m $$. A simplified model is shown in Figure 6. Therefore, the transfer function can be derived using the free-body diagram as follows:

Figure 5. Structure model parameter.

Figure 6. Diagram of a simplified structure model.

Absolute displacement of the robot end-effector, $$ x $$, is the sum of absolute structure displacement, $$ x_f $$, due to structure flexibility and relative displacement, $$ x_e $$, from the actuator movement.

Equations (3 and 4) are created by focusing on all forces applied to each mass; therefore, derived equations will get a transfer function, as shown in Equation (5).

(3) $$ \begin{eqnarray} m_f\ddot{x}_f &=& -d_f\ddot{x}_f - k_f x_f - u + d\dot{x}_e \end{eqnarray} $$

(4) $$ \begin{eqnarray} m\ddot{x} &=& u - d\dot{x}_e \end{eqnarray} $$

(5) $$ \begin{eqnarray} G_s &=& \frac{X_f}{X_e} = \frac{-ms^2}{(m_f + m)s^2 + d_f s + k_f } \end{eqnarray} $$

3.3. Spectrum analysis

Spectrum analysis is the process of analyzing a signal in the frequency domain, which reveals the frequencies and their associated magnitudes and phases that are present in the signal. This can be used to identify frequency components within a signal. In this study, Spectrum analysis is used for creating a Bode diagram of a system using input and output signals that are measured from the test system.

The fast Fourier transform algorithm is a widely used method for spectrum analysis that can calculate the discrete Fourier transform of a given signal. However, this algorithm is unsuitable for analyzing noise signals that are random and stochastic because it does not have well-defined frequencies. To analyze noise signals, power spectral analysis is a more suitable method. It can estimate the power of a signal at each frequency. But this method is still for analyzing a single signal. A cross-power spectral density, an extension of power spectral analysis, can give the result of total noise power spectral density in complex values at each frequency and is used to analyze the correlation between two signals.

The cross-correlation method is used to find the amplitude and frequency relationship between two signals. It can be expressed as Equation (6), where $$ E $$ is the expected value. For finite discrete-time signals, it can be mathematically expressed as Equation (7), where $$ N $$ is the length of the signals $$ x(n) $$ and $$ y(n) $$, and $$ k $$ ranges from $$ -(N-1) $$ to $$ (N-1) $$.

The cross-power spectral density is the Fourier transformation of cross-correlation as Equation (8). For finite discrete-time signals, it can be mathematically expressed as Equation (9). In this study, the $$\texttt{cpsd}$$ function in MATLAB was used to calculate the cross-power spectral density.

The transfer function can be estimated using the cross-power spectral density of the input and output signals, as shown in Equation (10), and the magnitude and angle of the complex number results will yield the transfer function amplitude and phase.

In this experiment, white noise is generated and used as the input of the system. and the output of the system consists of actuator displacement from the rotary encoder attached to the servo motor and structure displacement from the laser distance sensor, as shown in Figure 7. This data will be used to determine the actuator system and structure characteristics.

Figure 7. Spectrum analysis experiment setup.

Actuator displacement, $$ x_e $$, is regarded as an input to the structure transfer function, $$ G_s $$, with frame displacement, $$ x_f $$, as an output. Therefore, by fitting a curve using the Bode diagram shown in Figures 8 and 9, the characteristic parameter of a structure can be obtained.

Figure 8. Bode diagram of structure.

Figure 9. Phase diagram of structure.

The actuator system receives control system signal input, $$ x_r $$, and output displacement, $$ x_e $$. From the result, we can obtain parameters in the actuator system transfer function, $$ G_a $$, by a fit graph of spectrum analysis, as shown in Figures 10 and 11. And we also use the Pade approximation to represent the time delay in the transfer function.

Figure 10. Gain diagram of an actuator system.

Figure 11. Phase diagram of an actuator system.

Resulting constants were $$ \omega_a = 75 \, \mathrm{rad/s} $$, $$ \zeta_a = 0.707, \, L_a = 3.5 \, \mathrm{ms}, \, m_f = 45 \, \mathrm{kg} $$, $$ m = 30 \, \mathrm{kg}, \, d_f = 420 \, \mathrm{Ns/m} $$, and $$ k_f = 320000 \, \mathrm{N/m} $$, respectively.

4.1. Block diagram and transfer function of human-machine system

To analyze the stability of the experiment system, including the human operator, we need to consider the block diagram, as shown in FFigure 12. The human operator impedance, $$ H $$, will react to errors that result from the absolute end-effector position, $$ x $$, subtract from the operator-hand intended position, $$ x_d $$, and create force, $$ f_h $$, for the admittance model, where hand impedance is used for modeling human operators as in Equation (11). Each parameter will change according to the controlled object and posture of the operator^[18,19].

Figure 12. Experiment setup block diagram.

Due to the insufficient stiffness of the robot, structure displacement that is created by robot movement has to be considered, so the absolute end-effector position, $$ x $$, will consist of a summation of the relative actuator displacement, $$ x_e $$, and the absolute structure displacement, $$ x_f $$. Therefore, the position-control robot transfer function, $$ G_r $$, consists of the linear actuator system transfer function, $$ G_a $$, and the structure transfer function, $$ G_s $$.

As a result, we can express the overall transfer function, $$ G_{sys} $$, from the intended position of the operator, $$ x_d $$, to the end-effector position, $$ x $$, with Equation (12).

4.2. Stability margin

Phase margin is a method used to check the stability of the feedback system. It measures how much phase shift can be applied before the system becomes unstable. To check the stability of a feedback system, we have to determine the open loop transfer function of the system and then find the gain crossover frequency, which is the point where the gain is unity. If the phase at that point is greater than $$ -180^{\circ} $$, the system is considered stable, and more phase margin means a more stable system. The system transfer function is expressed as Equation (12), and the phase margin can be calculated from the open-loop transfer function $$ H G G_a (G_s+1) $$.

When the system is in contact with a human operator, it might become unstable. The harder the operator grabs the end-effector, the more the system will oscillate, and the impedance of the operator, $$ H $$, is greatly affected by grip strength. Therefore, in stability analysis, the stiffness of the impedance of the operator, $$ k_h $$, is a major component to be considered for stability.

In this section, we simulated system stability with changes in $$ M $$ and $$ D $$ in the admittance model to see how they affected overall system stability. The human impedance constants are $$ m_h = 1 \, \mathrm{kg} $$ and $$ d_h = 17 \, \mathrm{Ns/m} $$, respectively, and $$ k_h $$ varies between $$ 1-10000 \, \mathrm{N/m} $$, based on our previous study^[18,19]. The simulation in Figures 13-15 is performed by fixing the mass parameter in the admittance model, $$ M $$, and varying the damping coefficient in the admittance model, $$ D $$, from $$ 0.1-1000 \, \mathrm{Ns/m} $$. Figures 16-18, on the other hand, will be simulated by fixing the damping coefficient constant and varying the mass parameter from $$ 0.1-1000 \, \mathrm{kg} $$.

Figure 13. Stability analysis with mass parameter, $$ M $$, at $$ 1 \, \mathrm{kg} $$.

Figure 14. Stability analysis with mass parameter, $$ M $$, at $$ 5 \, \mathrm{kg} $$.

Figure 15. Stability analysis with mass parameter, $$ M $$, at $$ 10 \, \mathrm{kg} $$.

Figure 16. Stability analysis with damping coefficient, $$ D $$, at $$ 1 \, \mathrm{Ns/m} $$.

Figure 17. Stability analysis with damping coefficient, $$ D $$, at $$ 5 \, \mathrm{Ns/m} $$.

Figure 18. Stability analysis with damping coefficient, $$ D $$, at $$ 10 \, \mathrm{Ns/m} $$.

According to the results in Figure 13, the system is unstable until the damping coefficient exceeds around $$ 100 \, \mathrm{Ns/m} $$. Figures 14 and 15 show that changes in the damping coefficient around $$ 0.1-20 \, \mathrm{Ns/m} $$ have little effect on the stability region of the system. Although the damping coefficient has a significant impact on system stability when its value exceeds $$ 20 \, \mathrm{Ns/m} $$, in the human-machine system admittance model, $$ G $$, must remain high for light operation, so this parameter must be small. Therefore, we can consider that the damping coefficient has no effect on the stability region in this study.

Moving on to Figures 16-18, it is shown that changing the value of the damping parameter, $$ D $$, in the admittance model may not change much in the stability region. In a stable region, the mass parameter, $$ M $$, greatly affects the phase margin. In conclusion, the results from the simulation show that changing the mass parameter will have a significant effect on system stability compared to changing the damping coefficient, which has little effect. So, in later simulations, we will only concentrate on changes in the mass parameter.

Figures 16-18 also show that to maintain system stability, the stiffness of the human operator, $$ k_h $$, must not exceed about $$ 1000 \, \mathrm{N/m} $$, and the mass parameter of the admittance model should be greater than around $$ 5 \, \mathrm{kg} $$. Because of insufficient structure stiffness and dead time in the actuation system, the stability region is decreasing from the ideal system. This restriction in stiffness of the human operator, $$ k_h $$, is allowable, but for the mass parameter, $$ M $$, it is too large for a human to operate comfortably. Therefore, we should consider adding a compensator to the system that will expand the stability region.

5.1. Design

A feed-forward compensator is a control system component that is used to improve the performance of the system. As shown in Figure 19, the compensator is inserted in series with the control system. In this study, we tried to minimize the effect of structure characteristics by adding a simple compensator to the system. The compensator, $$ C_{ff} $$, as shown in Equation (13), consists of an inverted transfer function from relative actuator displacement, $$ x_e $$, to absolute end-effector displacement, $$ x $$, but we do not include the actuator transfer function, $$ G_a $$, in the compensator because of its unstable zero that comes from the Pade approximation of dead time. The idea of this compensator is to cancel out the characteristics of the structure that cause the system to become unstable in some conditions. Therefore, the overall transfer function includes the compensator of the feed-forward compensated system, $$ G_{xxr} $$, which can be expressed as Equation (14).

Figure 19. Feed-forward system block diagram.

In Figure 20, the Bode diagram shows the response of the overall robot transfer function, $$ G_{xxr} $$, of the original system and feed-forward compensated system. In the original system, it has an effect from the structure characteristics at around $$ 70 \, \mathrm{rad/s} $$ that can cause an oscillation when a human is in contact with it. And after we append a feed-forward compensator, it shows that it can cancel out the effect of the structure characteristics.

Figure 20. Bode diagram of the original and feed-forward compensated system.

5.2. Simulation

The compensator that is applied to the system is the inverse of the structure transfer function, so we have to consider the error of the structure parameter measurement because if the measurement is not exactly the same as the actual system, the designed compensator will not cancel out the structure transfer function. With this situation, we also need to consider that the actual system has a different natural frequency from the measurement, and the parameter that directly changes natural frequency is the stiffness of the structure and the mass of the structure. In this simulation, we consider the change in structure stiffness, $$ k_f $$. So, in this simulation, we use parameter $$ D = 1 \, \mathrm{Ns/m} $$ in the admittance model, and we tried to simulate stability with an actual system that has the exact same stiffness as the measurement, as shown in Figure 21, the simulation with an actual system that has stiffness $$ 20\% $$ less than the measurement, as shown in Figure 22, and the simulation with an actual system that has stiffness $$ 20\% $$ greater than the measurement, as shown in Figure 23.

Figure 21. Feed-forward system stability analysis with actual system structure stiffness at $$ 100\% $$.

Figure 22. Feed-forward system stability analysis with actual system structure stiffness at $$ 80\% $$.

Figure 23. Feed-forward system stability analysis with actual system structure stiffness at $$ 120\% $$.

From Figure 21, when the compensator perfectly cancels out the structure characteristic, the stability region is increased compared to the system without compensation shown in Figure 16. It shows that a feed-forward compensator allows us to use a smaller $$ M $$ parameter in the admittance model to around $$ 0.2 \, \mathrm{kg} $$, which is an improvement from a non-compensated system that can use $$ M = 5 \, \mathrm{kg} $$ or above. However, when the actual system parameter is not the same as the measurement, it causes a reduction in the stability region. As shown in Figures 22, 23, the allowable $$ M $$ parameter in the admittance model is around $$ 1 $$ and $$ 1.5 \, \mathrm{kg} $$, respectively, which is much greater than the perfectly compensated system, as shown in Figure 21.

To easily compare the stability region between an uncompensated and a feed-forward system, we display the stability boundary of both simulations in the same graph, as shown in Figure 24 for a perfectly compensated system and Figure 25 for a system considered to have a $$ +20\% $$ stiffness parameter measurement error. These two figures show that errors in stiffness parameters greatly affect the stability region of a feed-forward compensated system. Although in Figure 25, the stability region of a feed-forward system is reduced, it still has an advantage over the system without compensation.

Figure 24. Stability region boundary comparison between a non-compensated system and a feed-forward system at $$ 100\% \, k_f $$.

Figure 25. Stability region boundary comparison between a non-compensated system and a feed-forward system at $$ 120\% \, k_f $$.

5.3. Experiment

In this experiment, we use the admittance model parameter as follows: $$ M = 2 \, \mathrm{kg}, \, D = 1 \, \mathrm{Ns/m} $$, because with these values, a non-compensated system will be in an unstable region, but a system with a feed-forward compensator will be in a stable region even though we take stiffness parameter measurement errors into account, as shown in Figure 25, which we can use to check whether the stability region is improving according to the simulation or not.

The result in Figure 26 shows that appending a feed-forward compensator to the system can improve the stability of the system. At $$ t = 0 \, \mathrm{s} $$ to $$ t = 2.5 \, \mathrm{s} $$, the machine is not in contact with humans, so both systems can remain stable, but at $$ t = 2.5 \, \mathrm{s} $$ to $$ t = 6.5 \, \mathrm{s} $$, where the human operator is in contact with the machine, it is clearly shown that a non-compensated system faces an aggressive oscillation, but on the other hand, the feed-forward system can reduce the oscillation by a significant amount, finally after $$ t = 6.5 \, \mathrm{s} $$ where the human operator releases the contact, it shows that the feed-forward system can quickly eliminate the oscillation compared to a non-compensated system that still oscillates even though humans already release their grip.

Figure 26. Non-compensated system and feed-forward system stability experiment with $$ M = 2 \, \mathrm{kg}, \, D = 1 \, \mathrm{Ns/m} $$.

6.1. Design

A feedback compensator or feedback controller is a control system that will measure the output signal of the system and adjust the input signal to the system. Compared between the feed-forward and feedback compensators, both will improve the performance of the system, but the advantage of the feedback system is that it tends to have better robustness, which means it can maintain performance despite changes in the environment.

As shown in Figure 27, it is a feedback control system in this study, and we use acceleration as the feedback signal because directly measuring absolute structure displacement is not easy in a practice situation. Therefore, we use an accelerometer, which can easily be attached to the frame. The downside of an accelerometer is that it cannot accurately measure a low frequency, but it can be used in this study because we consider only the oscillation frequency range. In Figure 27, the block of $$ s^2 $$ represents the accelerometer to get the structure acceleration, $$ a_f $$, from the structure displacement, $$ x_f $$, and $$ C_{fb} $$ is a compensator transfer function.

Figure 27. Feedback system block diagram.

To get better robustness to the change in structure characteristics, we need to make sensitivity small around the resonance frequency. The sensitivity of this system can be calculated using Equation (15).

The experiment system with a feedback controller, which can be expressed in Figure 27, can be derived into Equation (16). To find the sensitivity of the system, we must derivate Equation $$ G_{xxr} $$ by the structure transfer function, as shown in Equation (17), $$ G_s $$, and then multiply with $$ \frac{G_s}{G_{xxr}} $$, as shown in Equation (18).

We need to design the compensator, $$ C_{fb} $$, that makes the sensitivity small and also stabilizes the feedback system. We can use the Youla parameterization method ^[20], which is a design technique for the stable plant. Considering the feedback system shown in Figure 28, the Youla parameterization can be expressed in terms of a transfer function as Equation (19), where $$ Q $$ is the transfer function from $$ r $$ to $$ u $$. Using this stable parameter, $$ Q $$, all stabilizing compensators, $$ C $$, are parameterized as Equation (20).

Figure 28. Fundamental Youla block diagram.

In Equation (18), $$ C_{fb} $$ can be replaced from Equation (20), and the plant of the system, $$ P $$, is equal to $$ G_a G_s s^2 $$, and we will get the result shown in Equation (21).

Similarly, $$ G_{xxr} $$ can be expressed with the parameter $$ Q $$ as Equation (22).

Equation (21) shows that if $$ Q $$ is expressed as Equation (23), the sensitivity of the system, $$ S $$, will be $$ 0 $$. Furthermore, from Equation (22), $$ G_{xxr} $$ will be just $$ G_a $$ with the same $$ Q $$; this means that the effect of the robot structure can be completely eliminated.

But in reality, we know that $$ G_a $$ has an unstable zero that cause by dead time and also that $$ \frac{1}{s^2} $$ is unstable itself. Therefore, the parameter $$ Q $$ in Equation (23) is not stable. So, $$ G_a $$ is divided into two parts, which consist of stable zero and unstable zero, as shown in Equation (24), and we only use $$ G_{a0} $$, as shown in Equation (25), which is the stable zero part. The result of $$ Q $$ is shown in Equation (26).

Because of unstable $$ \frac{1}{s^2} $$, we must add at least a $$ 2^{nd} $$ high-pass filter to make $$ Q $$ stable. But if we append a $$ 2^{nd} $$ high-pass filter, the compensator will have some DC gain, so we use a $$ 3^{rd} $$ high-pass filter to reject the offset or bias of the acceleration measurement. And also, we append a low-pass filter to reduce the noise effect and improve robust stability, resulting in $$ Q $$ that is shown in Equation (27).

Finally, we can replace $$ Q $$ in Equation (20) with Equation (27) to calculate the compensator transfer function, $$ C_{fb} $$, as shown in Equation (28).

Parameters $$ \omega_1 , \, \omega_2 $$ are the cutoff frequencies of the high-pass filter and low-pass filter, respectively, and they must be selected so that the sensitivity of the system is reduced around the peak sensitivity of $$ 80 \, \mathrm{rad/s} $$, as shown in Figure 29. Here, the cutoff frequencies are considered across the peak frequency, and we select $$ \omega_1 = 22 \, \mathrm{rad/s}, \ \omega_2 = 300 \, \mathrm{rad/s} $$, respectively.

Figure 29. Sensitivity of original and feedback compensated system.

In Figure 30, the Bode diagram shows the response of the overall robot transfer function, $$ G_{xxr} $$, of the original system and feedback compensated system. It shows that the feedback compensation can reduce the effect of structure characteristics that occur in the original system. And also, because the sensitivity of the system in that region is reduced, the system can withstand the change in structure characteristics, which means the robustness of the system is increased.

Figure 30. Bode diagram of original and feedback compensated system.

6.2. Simulation

The simulation is conducted using the same method as a feed-forward compensator simulation. Simulation with an actual system that has exactly the same stiffness as the measurement, as shown in Figure 31, the simulation with an actual system that has stiffness $$ 20\% $$ less than the measurement, as shown in Figure 32, and a simulation with an actual system that has stiffness $$ 20\% $$ greater than the measurement, as shown in Figure 33.

Figure 31. Feedback system stability analysis with actual system structure stiffness at $$ 100\% $$.

Figure 32. Feedback system stability analysis with actual system structure stiffness at $$ 80\% $$.

Figure 33. Feedback system stability analysis with actual system structure stiffness at $$ 120\% $$.

Figure 31 shows that the stability region is improved over a non-compensated system, but it is still smaller than the stability region achieved with feed-forward compensation, as shown in Figure 21. However, when the actual system structure stiffness decreases from measurement by $$ 20\% $$, the stability region slightly expands, as shown in Figure 32. On the other hand, as shown in Figure 33, the stability region still decreases when actual system stiffness increases from measurement by $$ 20\% $$, but it can remain stable compared to feed-forward in Figure 23.

In Figure 34, it is shown that when the compensator is perfectly canceling out structure characteristics in feed-forward compensation, it has an advantage in the stability region over the feedback system. But when the system is not perfectly compensated, as shown in Figure 35, it shows that the feedback system can remain stable with little change, but the stability region of the feed-forward system reduces significantly to the point that the feedback system has a greater stability region than the feed-forward system.

Figure 34. Stability region boundary comparison between a non-compensated system, a feed-forward system, and a feedback system at $$ 100\% \, k_f $$.

Figure 35. Stability region boundary comparison between a non-compensated system, a feed-forward system, and a feedback system at $$ 120\% \, k_f $$.

6.3. Experiment

In the experiment, we use the admittance model parameter as follows: $$ M = 2 \, \mathrm{kg}, \, D = 1 \, \mathrm{Ns/m} $$, same as the previous experiment, so we can compare the result with feed-forward compensation.

From Figure 36, it is clear that a feedback compensator can greatly improve the stability of the system compared to a non-compensated system. However, Figure 36 also shows that the feedback system has low-frequency movement when a human operator is in contact with the machine, and it causes movement when the human operator attempts to make the end-effector remain in the same position. The damping parameter, $$ D $$, of the admittance model should be larger than what we used in this experiment for precise movement operations ^[19]. We should consider changing the damping parameter according to the phase of the task; for example, during reaching movement, the damping, $$ D $$, should be small, while for precise movements, larger damping, $$ D $$, should be employed, which we will consider in future studies.

Figure 36. Non-compensated system and feedback system stability experiment with $$ M = 2 \, \mathrm{kg}, \, D = 1 \, \mathrm{Ns/m} $$.

And when comparing feed-forward compensation with feedback compensation, as shown in Figure 37, feedback compensation has a slight improvement in stability over feed-forward compensation in this condition, so we try another experiment with a smaller $$ M $$ parameter. The admittance model parameter is as follows: $$ M = 1 \, \mathrm{kg}, \, D = 1 \, \mathrm{Ns/m} $$, which will make the feed-forward system unstable, but the feedback system can remain stable according to Figure 35, and the result is shown in Figure 38.

Figure 37. Feed-forward system and feedback system stability experiment with $$ M = 2 \, \mathrm{kg}, \, D = 1 \, \mathrm{Ns/m} $$.

Figure 38. Feed-forward system and feedback system stability experiment $$ M = 1 \, \mathrm{kg}, \, D = 1 \, \mathrm{Ns/m} $$.

Figure 38 shows that in a feed-forward system, it cannot remain stable even when a human operator is not in contact with it, and it starts to aggressively oscillate when a human is in contact at $$ t = 2.5 \, \mathrm{s} $$ to $$ t = 6.5 \, \mathrm{s} $$. On the other hand, a feedback system can remain stable the whole time with a little oscillation when a human operator is in contact with them.

Figure 37 and Figure 38 show that when we change parameter $$ M $$ in the admittance model, the feedback-compensated system can remain stable with operator interaction force within $$ \pm 1 \mathrm{N} $$. Low-frequency force in the graph comes from the instability of the human-machine system, which is caused by the low damping parameter in the admittance model, $$ D $$. However, the high-frequency oscillation force is greatly reduced compared with a feed-forward system, which means the influence of insufficient structure stiffness is compensated with feedback.

Authors' contributions

Conducted the experiment, simulation, data acquisition, and data analysis: Songthumjitti N

Conceived and designed the study: Inaba T

Interpreted the results: Songthumjitti N, Inaba T

Availability of data and materials

Not applicable.

Financial support and sponsorship

None.

Conflicts of interest

All authors declared that there are no conflicts of interest.

Ethical approval and consent to participate

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Copyright

© The Author(s) 2023.